VII.-Contracted Multiplication of Decimals. Rule. To multiply two decimals together, so as to retain only a certain number of places in the product, without the trouble of finding the rest― invert the order of the figures of the multiplier, and write them under those of the multiplicand in such a way that what was the units figure of the multiplier may come under the last place of decimals, which is to be retained. Multiply as usual, with this exception, that each figure of the multiplier begins with the figure of the multiplicand which comes immediately over it, the figure next to that being only used to carry from (as in the subsequent example). Put the several lines directly under one another, instead of removing each one place to the left. * As it is almost impossible to make this rule clear in words, we subjoin an example at length. Ex. To multiply 147 3861 by *6457, retaining only three places of decimals. The second factor, written so as to show a unit's place, is 0.6457, and in reversing, the 0 must fall under the third decimal place of the other factor, thus:: с 1473861 75460 88432 5895 737 103 95.167 Multiplier reversed; units place 0 falling under third decimal 6 of the upper line. Multiplier 6; figure to begin with, 8, figure to carry from, 6. Six times 6 is 36, nearest ten, four tens, carry four. Six times 8 is 48, and 4 is 52, put down 2 and carry 5. The rest as usual. Multiplier 4; figure to begin with, 3; figure to carry from 8. Four times 8 is 32; nearest ten, three tens, carry three. Four times 3 is 12 and 3 is 15, &c. The rest as usual. Multiplier 5; figure to begin with, 7; figure to carry from, 3. Five times 3 is 15; nearest* ten, two tens, carry two. Five times 7 is 35 and 2 is 37, &c. The rest as usual. Multiplier 7; figure to begin with, 4; figure to carry from, 7. Seven times 7 is 49, nearest ten, five tens, carry five. Seven times 4 is 28 and 5 is 33, &c. The rest as usual. Add as usual, and mark off three places; (the number proposed) for decimals. The full product of 147 3861 and 6457 is 95 16720477, which in thousandths only is nearest to 95 167, our result. The following multiplications have the proper arrangement and result given. No decimal places means that the whole number of the result is required, without fractions. VIII.—Contracted Division of Decimals. Rule.-Proceed as usual, until the number of quotient figures remaining to be found does not exceed the number of figures in the divisor. Then, instead of annexing a cipher, or bringing a figure down from the dividend, cut off the last figure of the divisor; that is, do not employ it except to carry from, as in the last rule. See how often this abridged divisor is contained in the remainder; multiply, carrying from the figure cut off; find a new remainder; cut off another figure from the divisor, and repeat the process until all the figures of the divisor are cut off. When the abridged divisor is not contained in the remainder, cut off a second figure from the divisor, put a cipher in the quotient, and proceed. We subjoin a detailed example. To divide 1299494 by 9915206, as far as nine decimal places. The first quotient figure being a decimal, and there being seven places in the divisor, two quotient figures must be found by the usual method; after which, the process is explained. In this case, 15 is equally near to one ten and two tens. It is usual, and generally more correct, to take the higher of the two. 171 8 ⚫273 Cut off the 6, reserving it to carry from; 991 not contained in 710; cut off one more Carrying figure 9. Divis. 9, contained once in 16. Once 9 is 9; nearest ten, one ten, carry one. Once 9 is 9, and 1 is 10. No divisor, carrying figure 9. What number of times 9 will carry 6, or be most nearly 60? Seven times 9 is 63; put 7 in the quotient, and carry 6, which finishes the process. Divisor. 3.14159265 2.7182818 51.77717 9 *414487636 9 8 ⚫003663004 ⚫00121922 membering the rule for increasing the last figure. Thus 16482677263, to two places only, may be found from 1648373, by the rule exemplified above. 414487636 74 529 *67272804 When the divisor itself contains more places than are required in the quotient, as many places may be cut from the right as will make the two the same; and the dividend may be cut down in the same way until no more places are left than will give one figure in the quotient, to the abridged divisor, reSECTION 3.-Extraction of the Square Root. Examples of Surds and Irrational Quantities. I.-Extraction of the Square Root. The rule for this will be better understood by a detailed example than by any verbal explanation. Though the quantities operated upon are decimal, it is to be understood that a whole number be used in the same way. may For 5, for instance, is 5'0000, &c. The following contains the working of the rule at length for the extraction Both in multiplication and division, it is best to retain one more place than is absolutely required to be correct. 25 106) 719 1127) 636 8300 7889 11343) 41100 113466) 34029 by figure, as below. (5.6736 First period 32; nearest square, 25; root 5. Put 5 in the root, and subtract 25 from 32. Remainder 7; bring down next period, 19. Double 5 (10), which place in divisor. Cut off one figure from 719,-71. This contains the divisor 10 seven times; try 7, as follows: annex it to divisor, 107; multiply by it, 107 X 7 is 749: this is greater than 719: 7 will not do. Try* 6. Then 106 × 6, is 636-less than 719. Put 6 in the root and in the divisor, and subtract 636 from 719; remainder, 83. Bring down next period, 00; add 6 last found to 106, giving new divisor, 112. Cut one figure from 8300-830. This contains 112 seven times. Try 7, and 1127 X 7 is 7889. Put 7 in the root and in the divisor, and subtract 7889 from 8300. Remainder, 411. Bring down next period, 00; add 7 last found to 1127, giving new divisor 1134. Cut one figure from 41100, -4110. This contains 1134 three times; trial* no longer necessary. Put 3 in the root, annex 3 to divisor, giving 11343. Subtract 11343 × 3, or 34029. 707100 Remainder, 7071. Bring down last period, 00; add 3 last found to 11343, giving new divisor 11346. Cut one figure from 707100-70710. This contains 11346 six times. Put six in the root; annex 6 to divisor, giving 113466. Subtract 113466 × 6, or 680796. 680796 26304 Remainder, 26304; less than half of 113466, which shows that there is no occasion to change the last found 6 into 7, to have the nearest decimal of four places. The required root is therefore 5 6736; by which we mean, that though 32 19 has no exact square root, yet 5.6736, multiplied by itself, will give a result nearer to 32.19 than any other number with four decimal places. This we will try. Multiply the three successive fractions, 5.6735, 5·6736, 5.6737, each by itself, retaining five decimal places in the product. *This trial will rarely be necessary after the second step. So that having cut one figure from the increased remainder, the number of times which the divisor is therein contained may be written down on the right, and the whole divisor, thus altered, multiplied by its last figure. |